Solve (64a^24)^-frac{1{6}}

$$( 64 a ^ { 24 } ) ^ { - \frac { 1 } { 6 } }$$

Evaluate

$$\frac{1}{2a^{4}}$$

Solution Steps

Expand \left(64a^{24}\right)^{-\frac{1}{6}}.

$$64^{-\frac{1}{6}}\left(a^{24}\right)^{-\frac{1}{6}}$$

To raise a power to another power, multiply the exponents. Multiply 24 and -\frac{1}{6} to get -4.

$$64^{-\frac{1}{6}}a^{-4}$$

Calculate 64 to the power of -\frac{1}{6} and get \frac{1}{2}.

$$\frac{1}{2}a^{-4}$$

Differentiate w.r.t. a

$$-\frac{2}{a^{5}}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\frac{1}{6}\times \left(64a^{24}\right)^{-\frac{1}{6}-1}\frac{\mathrm{d}}{\mathrm{d}a}(64a^{24})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-\frac{1}{6}\times \left(64a^{24}\right)^{-\frac{7}{6}}\times 24\times 64a^{24-1}$$

Simplify.

$$-256a^{23}\times \left(64a^{24}\right)^{-\frac{7}{6}}$$